Saturday 7th October 2017
In the mornings, I (Robert) am currently reading parts of two long histories of England simultaneously, while I eat my breakfast cereal. I had long thought of doing this what you might call "stereo reading" in two books at once (in practice, I read a certain distance in one of them, then read how the same particular period-topic is treated in the other). It's a powerful aid to the memory and the understanding and a spur to one's interest: to see how two different historians describe the same reality. I've picked 1714 as my starting point, since I find the eighteenth century especially mysterious.
One of the books is G M Trevelyan's "History of England". The other is Keith Feiling's. Both are massive old tomes, written in an age when historians could write well while also writing in a stately formal style. But even so, they are interestingly different from one another.
Wednesday 4th October 2017
An A to Z of some mathematical terms
Arc an arc is part of the circumference of a circle or of any other curve.
Bar chart a chart using bars to represent quantities.
Chord a chord is a straight line joining two points on a circle.
Denominator the number below the line in a fraction.
Equilateral a triangle which has all its sides equal in length.
Factorial the product of a number and all the numbers below it, for example, 5 factorial (written as 5!) = 5x4x3x2x1.
So, 10! = 10x9x8x7x6x5x4x3x2x1
Geometric Series a sequence of numbers in which each is obtained by multiplying the previous number by a given amount, the amount being the same each time. For example, the sequence 1, 2, 4, 8, 16, 32 where each number is multiplied by 2.
Or 100, 50, 25, 12.5, 6.25, 3.125 where each number is multiplied by a half.
Hypotenuse the longest side in a right-angled triangle.
i i is an imaginary number and is the square root of -1. Complex numbers use i. By contrast, a real number cannot have a negative square root.
Kite a kite is a quadrilateral figure which is symmetrical about one of the diagonals. It is made up of two isosceles triangles which have a common base.
Linear equation an equation between two variable terms that gives a straight line when plotted on a graph. For example, y=x and y=3x+1.
Matrix a rectangular array of numbers or letters in rows and columns.
Node a point at which lines or curves intersect.
Obtuse angle an angle which is greater than 90 degrees and less than 180 degrees.
Product a result obtained by multiplying quantities together.
Quadrant a quarter of a circle, bounded by two radii at right angles and the arc cut off by them.
Reflex angle an angle which is greater 180 degrees.
Stationary point a point on a curve where the gradient is zero.
Trapezium a quadrilateral with just one pair of parallel sides.
Unity the number 'one'; the factor that leaves unchanged the quantity on which it operates.
Variable a quantity that can assume different numerical values.
Whole number a number without a fraction part; an integer.
Zero 0; the point from which positive or negative quantities are reckoned.
Monday 2nd October 2017
Logarithms at A Level are, from my experience, not an easy topic. I find that I need to go back to basics if I haven't done any for a while. Here is an attempt at explaining them.
A logarithm is another word for an index or power.
3 2 = 9
Here, 2 is the logarithm which, with a base of 3, gives 9. In log notation, the base is written as a subscript after the word 'log', thus for base 3: log 3
The 9 goes next: log 3 9
and lastly, we complete the equation with the logarithm, or power, 2: log 3 9 = 2
This says, 2 is the logarithm, or power, to which 3 is raised to give 9.
Another example, 4 3 = 64
Writing this in log notation, we have a base of 4 and a logarithm of 3: log 4 <something> = 3
The <something> is the number we haven't used,
i.e. 64: log 4 64 = 3
'The power to which 4 needs to be raised to give 64 is 3'.
Monday May 15th 2017
Area of a Triangle
There are two formulae to find the area of a triangle. If you know the perpendicular height of the triangle, you can use the formula 1/2 x b x h, where b is the length of the base and h is the height of the triangle. The second formula is 1/2 x a x b x sin C, a and b being the lengths of two sides and C being the angle in between these two sides. If you have the lengths of all three sides but no angle, then you can use the cosine formula to find one angle and then you can use 1/2 x a x b x sin C.
To find C, use this arrangement of the cosine formula:-
cos C = (a² + b² - c²)/2ab
Wednesday March 29th 2017
Abundant Numbers - Weird Numbers - Deficient Numbers
All numbers have factors - numbers that divide into them. The factors of 10 are 1, 2, 5 and 10. For this discussion, we will leave out the number itself, that is, for 10 we are just considering the factors 1, 2 and 5. 1 + 2 + 5 = 8 which is less than 10 and so it is called a deficient number.
An abundant number has factors which add up to more than the number. The first abundant number is 12. Its factors add up as follows: 1 + 2 + 3 + 4 + 6 = 16 making it abundant as it is more than 12.
A weird number is a special sort of abundant number. The number 12 is not a weird number because you can make a total of 12 from some of its factors, thus: 2 + 4 + 6 = 12. The smallest weird number is 70 as you can't find a set of factors which total 70. Its factors are: 1, 2, 5, 7, 10, 14 and 35, adding up to 74, making 70 abundant.... but weird!
Weird numbers are rare - these are the only ones below 10,000: 70, 836, 4030, 5830, 7192, 7912 and 9272.
Tuesday March 28th 2017
Welcome to our blog of helpful information for English, ESL (English as a Second Language), French and Mathematics.
We begin with regular -er verbs in French. A good verb to start with is 'parler' which means 'to speak'. It's useful to learn the endings of the verb, which in the case of regular -er verbs are:
-e, -es, -e, -ons, -ez, -ent
Just taking the masculine form of the verb, we have:
je parle I speak
tu parles you speak (familiar)
il parle he speaks
nous parlons we speak
vous parlez you speak (formal or plural)
ils parlent they speak
This is how we learnt at school in the sixties and seventies and we believe we were taught well. Just learn those endings : -e, -es, -e, -ons, -ez, -ont. Say them aloud until you know them off by heart.
Then learn the whole pattern - we call it a conjugation - so, je parle, tu parles, il parle, nous parlons, vous parlez, ils parlent. Say it out loud. It's a great way to learn!
When you know the basics well, you can add in:
'elle parle' after 'il parle'
'elles parlent' after 'ils parlent'